Shifting paths to avoidable ones

An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milani\v{c}, and Servatius conjectured that every graph that contains an induced $k$-vertex path also contains an avoidable induced path of the same length, and proved the result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced $k$-vertex paths are shifts of each other if their union is an induced path with $k+1$ vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.